THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

12.12 Noise Reduction of a Wall 303Combining the last two equations gives the total energy δ 2 density near the wall:( 1δ 2 = + 4 )W2S w R 2 c . (12.31)Equation (12.28) can now be used to eliminate W 2 in Equation (12.31) to yieldδ 2 = 4 (W 1 1R 1 c 4 + S )wτ. (12.32)R 2The mean-square pressure in room 2 is given in terms of energy density byp2 2 = ρ 0c 2 δ 2which is then inserted into Equation (12.32) to yieldp2 2 = 4W (1 1ρ 0 cτR 1 4 + S )w. (12.33)R 2From the use of the definitions( ) 2 ( )p2W1L p2 = 10 logand L W 1 = 10 log20 μPa10 pWEquation (12.33) becomes( ) ( ) ( 4 1 1L p2 = L w1 + 10 log − 10 log + 10 logR 1 τ 4 + S )w(12.34)R 2where we have assumed that ρ o c = 407 rayls.In inspecting Equation (12.34) we observe that the first two terms in the righthandside of the equation represents L p1 under the conditions of a reverberant fieldnear the wall in room 1. We also invoke Equation (12.3) which expresses TL interms of transmission coefficient τ, with the result that Equation (12.34) simplifiesto:( 1L p2 = L p1 − TL + 10 log4 + S )w. (12.35)R 2From Equation (12.35) one can estimate the sound pressure level L p2 near the wallin room 2, given the TL of the wall and the acoustic parameters of room 2. On theother hand, if we know the desired value of L p2 , (which is the usual case) we canrearrange Equation (12.35) to find the necessary transmission loss of the wall asfollows:( 1TL = L p1 − L p2 + 10 log4 + S )w. (12.36)R 2The above equation is valid in both English and metric units.The term L p1 − L p2 is referred to as the noise reduction denoted by the termNR. Equation (12.36) becomes( 1NR=TL− 10 log4 + S wR 2). (12.37)